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Weighted differentiation composition operators from mixed-norm spaces to weighted-type spaces. (English) Zbl 1165.30029
Summary: Motivated by the recent paper [{\it X. Zhu}, Integral Transform. Spec. Funct. 18, No. 3, 223--231 (2007; Zbl 1119.47035)], we study the boundedness and compactness of the weighted differentiation composition operator $D_{\varphi,u}^n (f)(z)=u(z)f^{(n)}(\varphi(z))$, where $u$ is a holomorphic function on the unit disk $\Bbb D, \varphi$ is a holomorphic self-map of $\Bbb D$ and $n\in \Bbb N_0$, from the mixed-norm space $H(p, q, \phi)$, where $p,q > 0$ and $\phi$ is normal, to the weighted space $H_\mu^\infty$ or the little weighted space $H^\infty_{\mu,0}$. For the case of the weighted Bergman space $A_\alpha^p, p > 1$, some bounds for the essential norm of the operator are also given.

##### MSC:
 30H05 Bounded analytic functions
Full Text:
##### References:
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